To find the quotient of [tex]\(2 \frac{2}{3} \div \frac{1}{4}\)[/tex], follow these steps:
1. Convert the mixed number to an improper fraction:
The mixed number [tex]\(2 \frac{2}{3}\)[/tex] can be converted to an improper fraction. To do this, multiply the whole number part by the denominator of the fractional part, then add the numerator:
[tex]\[
2 \frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3}
\][/tex]
2. Set up the division problem:
Now, we need to divide [tex]\(\frac{8}{3}\)[/tex] by [tex]\(\frac{1}{4}\)[/tex].
3. Convert the division to multiplication using the reciprocal:
When you divide by a fraction, you can multiply by its reciprocal. The reciprocal of [tex]\(\frac{1}{4}\)[/tex] is [tex]\(\frac{4}{1}\)[/tex]:
[tex]\[
\frac{8}{3} \div \frac{1}{4} = \frac{8}{3} \times \frac{4}{1}
\][/tex]
4. Multiply the fractions:
Multiply the numerators together and the denominators together:
[tex]\[
\frac{8 \times 4}{3 \times 1} = \frac{32}{3}
\][/tex]
5. Convert the improper fraction back to a mixed number:
Divide the numerator by the denominator to find the whole number part, and the remainder will be the numerator of the fractional part:
[tex]\[
32 \div 3 = 10 \text{ remainder } 2
\][/tex]
Thus,
[tex]\[
\frac{32}{3} = 10 \frac{2}{3}
\][/tex]
Therefore, the quotient of [tex]\(2 \frac{2}{3} \div \frac{1}{4}\)[/tex] is [tex]\(10 \frac{2}{3}\)[/tex]. The whole number part of the mixed number is [tex]\(10\)[/tex], which is the answer in simplest form.
